The LGA is calling on the government to remove restrictions which mean a school must have academy status before it can sponsor a school in need of assistance, Physorg reported.
The LGA says this would allow councils to intervene much more swiftly when parents raise concerns about a school.
Ministers say becoming an academy is the best way to improve standards.
But David Simmonds, chairman of the LGA's Children and Young People Board, said 80 percent of maintained schools were rated as good or outstanding by Ofsted inspectors.
These schools were, therefore, in a strong position to help other council controlled schools which were struggling, he said.
He noted that when parents had concerns about standards in a non-academy school, they turned to the local authority for help.
"We could intervene much more swiftly, but there's a requirement that schools convert to academy status before being able to sponsor a struggling school now that takes time.
"It's a piece of bureaucracy that doesn't need to be there."
Simmonds said the Education and Adoption Bill, which has been presented to and debated in the Commons and is now being examined by the Public Bill Committee, would be an opportunity to remove the restriction.
Most primary schools in England are still overseen by local authorities
"For parents, who are far more concerned with the quality of their child's education in the classroom than the legal status of the school, it is the council that they still turn to for advice and support.
"However, their current powers to intervene are strictly limited.
"The Education and Adoption Bill provides the ideal opportunity to right this wrong and should allow councils and the best maintained schools to share expertise and provide support to school leadership teams to ensure standards rapidly improve."
But the government says the bill will "remove bureaucratic and legal loopholes" that slow down the process of turning failing schools themselves into academies.
A spokesman for the Department for Education said: "We are clear that the most effective way of bringing about rapid and sustained improvement is to become an academy with the support of a strong sponsor.
"All good and outstanding schools have the freedom to convert to academy status either as standalone converters or as part of a multi-academy trust to share their expertise with other schools.
"Academies are a vital part of our plan for education and the advantages and freedoms enjoyed by these schools are transforming the education of millions of pupils across the country."
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